Effect of anisotropy pores on thermohaline convective instability in micropolar ferrofluid

This work deals with the thermohaline convective instability on horizontal micropolar ferrofluid layer in anisotropic porous effect subjected to a transverse uniform magnetic field. The linear stability analysis is used to omit the non-linear terms on governing equations, normal mode analysis is taken to the study and free boundary conditions are applied. The critical magnetic thermal Rayleigh number N SC is calculated analytically for sufficient large values of M 1 The principle of exchange of stabilities is satisfied for micropolar ferrofluid in an anisotropic porous medium without N 1, N ’ 3, N ’ 5 and τ. The sufficient conditions for the non-existent of overstability are also examined. Moreover, in this work, we tried to investigate the anisotropy effect on porous medium on the system. The parameters N 1and N 5 ’ dominate the system and the large porous medium is taken into account.


Introduction
The double diffusion related to the thermohaline convective effects were first discovered an outstanding solutions of the various investigations by Sterm [1], Turner [2]- [3], Huppert and Turner [4] - [5] and Turner and Stommel [6]. The phenomena on double diffusive convective instability excellently investigated by Turner [6]. Turner and Chen [7] have taken to analyse the sugar-salt system and shadowgraph photography is made to excellent work. Veronis [8] investigated the thermohaline convective instability in fluid layer. This thermohaline convective instability has been obtained by Veronis [9] in the in the Boussinesq approximation. Huppert and Moore [10] have obtained a various results of non-linearity on double diffusive convetion and then Knobloch et al. [11] used a five mode truncation given by Veronis [8] to get the solution that were in excellent qualitative agreement with the numeral results of Huppert and Moore [10]. The two illustrations of 2D non-linear double diffusive convective instability has been studied by Knobloch and Proctor [12]. In this, non-linear solution could be found analytically. The unstable mode has been studied in the most cases on thermohaline convective instability subjected to linear grandients by Baines and Gill [13] and they introduced salinity Rayleigh number. Vaidyanathan et al. [14] examined the ferrothermohaline convection.
The flow of fluid phenomena with porous behavior is a important investigation because of its natural case. The stability analysis of fluid flow in a porous behavior was taken by Wooding [16] and Lapwood [15] with the Darcy resistance. Moreover, the effect of porous on double diffusive convective instability in a fluid is of interesting study in engineering sciences. Nield and Bejan [17] gave the double diffusive convective instability in porous medium. The engrossing features of the ferrofluid is the hope of influencing flow by the magnetic field and vice-versa (Feynman et al. [19], Shliomis [20]). High quality review on the convection of ferrofluid has been done by Rosensweig [21]. Finlayson [22] opened the convective instability work on ferrofluid layer with uniform vertical magnetic field. Vaidyanathan et al. [23] studied the porous effect on Finlayson [22] using Brinkman number. Then, this analysis was taken for investigate with the anisotropy effect on Brinkman pores by Sekar et al. [24].
Eringen [25] introduced the micropolar fluids theory. This theory has been developed by Eringen [26] on thermal effect. An excellent reviews and applications of this fluids theory can be obtained in by Ariman et al. [27] and Eringen [28]. Rosensweig [21] suggested the wonderful monograph on ferrofluid. In this, it is apt to take the microrotation behavior on the particles. Due to this truth, the works have been assumed by treating the ferrofluid as micropolar fluids. The Rayleigh-Bénard convection problem has been analyzed by Abraham [29] on micropolar ferrofluid layer. In this, it is allowed by uniform magnetic field and stress-free boundaries is taken. Sunil and Bharti [30] has been undertaken the porous effect in micropolar ferrofluid. The thermosolutal convection has been taken by Sharma and Sharma [31] in micropolar fluids in an existence of a porous medium. Sunil et al. [32]- [38] investigated the double diffusive convective instability in micropolar ferrofluid in an existence of pores and non-pores effects.
Our plan is prolong this work to the investigation of thermohaline convective instability in Eringen's micropolar fluid in an existence of anisotropic porous effect and uniform magnetic field taken into account. In other words, an infinite horizontal micropolar ferrofluid layer heated from below and it is salted from above existence of anisotropic porous medium. The thickness of the fluid layer is and the temperature and salinity at the bottom and top surfaces are and respectively and and are maintained.

Mathematical formulation of problem
The continuity equation is (1) The momentum equation is ( ) (2) The internal angular momentum equation is ( ) The basic state quantities obtained are A small thermal disturbance is made on the system. Let us take the perturbed components of M and H be and respectively. The perturbed state quantities are (9) The equation (7) can be calculated as (10) Then the x, y and z components of Eq. (2) become Eqs. (3) can be calculated as ( )

Normal mode analysis technique
We undertake the perturbed quantities by use of normal modes are (17) where represents The Salinity equation is

Linear stability theory
We consider the free boundary conditions. The boundary conditions on , , and are (28) The exact solutions satisfying above Eq.
The determinant of co-efficients of A, B, C, E and F must vanish for the existence of non-trivial Eigen functions. Eqs. (29)- (33) have been adopted to obtain (35) This gives that the porous medium, anisotropy parameter and Salinity Rayleigh number always have a destabilizing behavior, if for stationary convection. In the non-presence of coupling parameter, N , the permeability of medium, anisotropic effect and stable solute gradient have a destabilizing behavior and also the same behavior is shows for the absence of spin diffusion effect (

Principle of exchange of stabilities
We analyze the possibility of oscillatory modes, if any, on stability problem due to the presence of magnetic numbers, porous medium, micropolar parameters, anisotropy porous medium and salinity gradient. Then, equate the imaginary part of Eq. (35), we obtain { } (49) where It is very clear from Eq. (49) that almost either non-zero or zero which means that the modes almost either oscillatory or non-oscillatory instabilities. In the non-presence of N , and , we obtain the result as In Eq. (50), is positive definite. Therefore, it gives that oscillatory modes are not permitted and the principle of exchange of stabilities is hold for micropolar ferrofluid layer heated from below and salted from above saturating an anisotropy and , which were non-existent in their non-presence.

The case of overstability
In this part, we examine the possibility that the observed instability may really be overstability. Since we desire to examine through the state of good oscillations, it is adequate to obtain the conditions for which Eq. (35) will allow solutions with real Then, we equate real and imaginary part of Eq.
The coefficients B and B of Eq. (51) are lengthy structure and it is not required in the examination of overstability.

Results and Discussion
The classical linear stability analysis is taken to analyse the thermohaline convective instability on micropolar ferrofluid with uniform angular velocity. The anisotropic effect is considered on porous medium. The thermal perturbation method and normal mode technique are used to get the solution. The stationary and oscillatory instabilities are obtained. In this investigation, we tried to analyse the effect of anisotropy porous on thermohaline convective instability in micropolar ferrofluid and Brinkman method is considered. Before we analyze the various physical quantities, we first form some physical comments on these like M is taken to be 1000 and M is assumed to be zero, M is ranges from 5 to 25 (Vaidyanathan et al. [18]). The porous medium k is ranges from 0.1 to 0.9. The anisotropic porous medium assumed from 0.3 to 3.1 (Sekar et al. [24]) and is taken as 0.05 (0.02) 0.11 (Vaidyanathan et al. [18]). R is ranges from -500 to 500 and M and M are taken to be 0.1 and M = 0.5. Moreover, N , and are getting some physical comments due to the suspended particles. Assuming the Clausius-Duhem inequality, Eringen [34] given the non-negativeness of N , and It is clear that the couple stress comes into play at small values of This supports the condition that and that is small positive real number and has to be positive finite number (Sunil et al. [33]). Fig. 1 shows the variation of N versus N for various k and anisotropic porous medium . This gives that k and have stabilizing effect. When the layer is taken to be following in an anisotropic porous medium, then an anisotropic porous effect has a destabilizing behavior and this behavior gets for k also. This is because, as anisotropy effect and k increases, the void space increases and the fluid flow gets on the plane will be increased clearly. Naturally, isotropic and anisotropic porous medium have a destabilizing behavior which was investigated by (Sekar et al. [35]). Fig. 2 displays the variation of N versus N for various R . It is obvious from the Fig. 2 that coupling parameter N has a stabilizing behavior on the system for increasing of R from -500 to 0 and system gets high energy. But, an influence of R (= 100 and 500) the system gets null effect. In other words, the convective system has an equilibrium state. When increasing value of salt on the fluid layer, the fluid is released to the lowest viscosity. Due to this, convection of the fluid is lead to fast. Fig. 3 represents the plot of N versus N for various M . This figure shows clearly that N has stabilizing effect for increasing of M . When M = 5, the system gets high energy due to the low magnetic field and noticed that the increasing of M from 10 to 25, the system observed low energy due to its highest magnetic field. Figs. 1-3 have analyzed for increasing values of from 0.05 to 0.11.
Figs. 4 and 5 show the variation of N with respect to , k and . In these figures, it is very clear that has destabilizing behavior. It gives that increases with increasing of permeability of medium k , anisotropy parameter N decreases. In Fig.  4, when increasing of k from 0.1 to 0.9, thermal energy decreases and it is more on the system when k = 0.9. Moreover, this convection process is just opposite to an anisotropy parameter. When increased from 0.7 to 3.1, N close to zero. In this moment, the system is promoted to salt. Likewise, when R = 100 and N gets slightly decreased with high thermal energy. But, and 0.09, the system leads to rest, which is depicted in Fig. 5.
In Fig. 6 Figs. 7-11 give the variation of N with respect to for various k , and R , respectively. Fig. 7 and 8 show that the heat induced into the fluid due to microelements is increased when increases. Thus increasing of gives to increase in N . Hence, has always a stabilizing flow for M = 5 and 10. Further, it is observed that Fig. 7 has exponential increase for Where as Fig. 8 has exponential increase for all Figs. 9 and 10 analyzed for , R = -500 and R = 500, respectively. The nature of the stabilizing behavior is made for presence and absence of salt on the system. When R = -500, N gets a highest value also the same effect is made for R = 500, but at this moment, the system has a low energy. Fig. 11 is illustrated for M = 20 for various R . In this situation, when R is increasing from -500 to 0, N is increased. Therefore, the system gets stabilizing flow. But, for R = 100 and 500, the system has internal energy due to heavy salting on the system. In Fig. 12, the variation of N versus is analyzed in existence and non-existence of coupling parameter N . It is clear that anisotropy effect has destabilizing behavior. This is indicated by a decrease in N , which is given by Sekar et al. [35] in non-existence of N . In existence of N (= 0.2), convective system gets high energy, but in non-existence of N (= 0), the convective system gets low energy. However, N converges to zero when Hence, the system has an equilibrium state. Fig. 13 gives the variation of N versus M for different in existence and non-existence of N . It is seen from this figure that as M increases from 5 to 25, N decreases. The system gets destabilized effect even with and without N .